Properties

Label 18.18.a.c
Level $18$
Weight $18$
Character orbit 18.a
Self dual yes
Analytic conductor $32.980$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [18,18,Mod(1,18)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(18, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 18, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("18.1");
 
S:= CuspForms(chi, 18);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 18 = 2 \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 18.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(32.9799757220\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 6)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + 256 q^{2} + 65536 q^{4} - 645150 q^{5} + 3974432 q^{7} + 16777216 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 256 q^{2} + 65536 q^{4} - 645150 q^{5} + 3974432 q^{7} + 16777216 q^{8} - 165158400 q^{10} + 500068668 q^{11} - 5425661314 q^{13} + 1017454592 q^{14} + 4294967296 q^{16} + 5466992958 q^{17} - 53889877060 q^{19} - 42280550400 q^{20} + 128017579008 q^{22} - 578906836536 q^{23} - 346720930625 q^{25} - 1388969296384 q^{26} + 260468375552 q^{28} + 4619583681690 q^{29} - 6802815567448 q^{31} + 1099511627776 q^{32} + 1399550197248 q^{34} - 2564104804800 q^{35} - 19571909422138 q^{37} - 13795808527360 q^{38} - 10823820902400 q^{40} - 57213620756922 q^{41} - 24501250225084 q^{43} + 32772500226048 q^{44} - 148200150153216 q^{46} - 184283998832832 q^{47} - 216834404264583 q^{49} - 88760558240000 q^{50} - 355576139874304 q^{52} + 206542562280354 q^{53} - 322619301160200 q^{55} + 66679904141312 q^{56} + 11\!\cdots\!40 q^{58}+ \cdots - 55\!\cdots\!48 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
256.000 0 65536.0 −645150. 0 3.97443e6 1.67772e7 0 −1.65158e8
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 18.18.a.c 1
3.b odd 2 1 6.18.a.a 1
12.b even 2 1 48.18.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.18.a.a 1 3.b odd 2 1
18.18.a.c 1 1.a even 1 1 trivial
48.18.a.f 1 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 645150 \) acting on \(S_{18}^{\mathrm{new}}(\Gamma_0(18))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 256 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T + 645150 \) Copy content Toggle raw display
$7$ \( T - 3974432 \) Copy content Toggle raw display
$11$ \( T - 500068668 \) Copy content Toggle raw display
$13$ \( T + 5425661314 \) Copy content Toggle raw display
$17$ \( T - 5466992958 \) Copy content Toggle raw display
$19$ \( T + 53889877060 \) Copy content Toggle raw display
$23$ \( T + 578906836536 \) Copy content Toggle raw display
$29$ \( T - 4619583681690 \) Copy content Toggle raw display
$31$ \( T + 6802815567448 \) Copy content Toggle raw display
$37$ \( T + 19571909422138 \) Copy content Toggle raw display
$41$ \( T + 57213620756922 \) Copy content Toggle raw display
$43$ \( T + 24501250225084 \) Copy content Toggle raw display
$47$ \( T + 184283998832832 \) Copy content Toggle raw display
$53$ \( T - 206542562280354 \) Copy content Toggle raw display
$59$ \( T - 418648048246140 \) Copy content Toggle raw display
$61$ \( T - 2501287878088382 \) Copy content Toggle raw display
$67$ \( T + 145692866050948 \) Copy content Toggle raw display
$71$ \( T - 5364313152664248 \) Copy content Toggle raw display
$73$ \( T - 3302058927938186 \) Copy content Toggle raw display
$79$ \( T - 22\!\cdots\!60 \) Copy content Toggle raw display
$83$ \( T + 20\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( T - 56\!\cdots\!10 \) Copy content Toggle raw display
$97$ \( T + 11\!\cdots\!18 \) Copy content Toggle raw display
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